The expression $3 (sin x - cos x)^{4} + 6 (sin x + cos x)^{2} + 4 ({sin}^{6} x + {cos}^{6} x)$ is equal to

10

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11

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12

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13

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Solution

The correct option is D $13$
Given expression $= 3 [(sin x - cos x)^{2}]^{2} + 6 (sin x + cos x)^{2} + 4 ({sin}^{6} x + {cos}^{6} x)$

$= 3 [{sin}^{2} x + {cos}^{2} x - 2 sin x cos x]^{2} + 6 [{sin}^{2} x + {cos}^{2} x + 2 sin x cos x] + 4 ([{sin}^{2} x]^{3} + {[{cos}^{2} x]}^{3})$

$= 3 [1 - sin 2 x]^{2} + 6 [1 + sin 2 x] + 4 \times [({sin}^{2} x + {cos}^{2} x)^{3} - 3 {sin}^{2} x {cos}^{2} x ({sin}^{2} x + {cos}^{2} x)]$

$= 3 [1 + {sin}^{2} 2 x - 2 sin 2 x] + 6 + 6 sin 2 x$ $+ (4 - 3 {sin}^{2} 2 x)$

$= 3 + 3 {sin}^{2} 2 x - 6 sin 2 x + 6 + 6 sin 2 x$ $+ 4 - 3 {sin}^{2} 2 x$

$= 13$

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Similar questions

3 (sin x - c o s x)^{4} + 6 (s i n x + c o s x)^{2} + 4 (s i n^{6} x + c o s^{6} x)

3 (sin x - cos x)^{4} + 4 ({sin}^{6} x + {cos}^{6} x) + 6 (sin x + cos x)^{2}

3 (sin x - cos x)^{4} + 6 (sin x + cos x)^{2} + 4 ({sin}^{6} x + {cos}^{6} x)

3 (s i n x - c o s x)^{4} + 6 (s i n x + c o s x)^{2} + 4 (s i n^{6} x + c o s^{6} x)

3 {(sin x - cos x)}^{4} + 4 ({sin}^{6} x + {cos}^{6} x) + 6 {(sin x + cos x)}^{2} = 13

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